Copyright © 2010 You-You Chen and Li-Xin Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let {U1,U2,…,Un} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process as αn(t)=n−1/2∑i=1n(I{Ui≤t}−t),0≤t≤1, ‖αn‖=sup0≤t≤1|αn(t)|. In this paper, we get the exact convergence rates of weighted infinite series of E‖αn‖2I{‖αn‖≥ε(logn)1/β}.